On a Fixed Point for Generalized Contractions in Generalized Metric Spaces

Abstract

Lakzian and Samet 2010 studied some fixed-point results in generalized metric spaces in the sense of Branciari. In this paper, we study the existence of fixed-point results of mappings satisfying generalized weak contractive conditions in the framework of a generalized metric space in sense of Branciari. Our results modify and generalize the results of Laksian and Samet, as well as, our results generalize several well-known comparable results in the literature. Definition 1.1 see 1 . Let X be a nonempty set and d : X × X → 0, ∞ such that for all x, y ∈ X and for all distinct points u, v ∈ X each of them different from x and y, one has 2 Abstract and Applied Analysis p1 :Then, X, d is called a generalized metric space or shortly g.m.s . Any metric space is a generalized metric space, but the converse is not true 1 .Definition 1.2 see 1 . Let X, d be a g.m.s, {x n } a sequence in X, and x ∈ X. We say that {x n } is g.m.s convergent to x if and only if d x n , x → 0 as n → ∞. We denote this by x n → x. Definition 1.3 see 1 . Let X, d be a g.m.s and {x n } a sequence in X. We say that {x n } is a g.m.s Cauchy sequence if and only if for each ε > 0 there exists a natural number N such that d x n , x m < ε for all n > m > N. Definition 1.4 see 1 . Let X, d be a g.m.s. Then, X, d is called a complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in X. Very recently, Lakzian and Samet 9 proved the following nice result. Theorem 1.5. Let X, d be a Hausdorff and complete generalized metric space. Suppose that T : X → X is such that for all x, y ∈ X ψ d Tx, Ty ≤ ψ d x, y − φ d x, y , 1.1where ψ : 0, ∞ → 0, ∞ is continuous and nondecreasing with ψ t 0 if and only if t 0, and φ : 0, ∞ → 0, ∞ is continuous and φ t 0 if and only if t 0. Then, there exists a unique point u ∈ X such that u Tu.

Periodic and fixed point using weaker Meir-Keeler function in complete generalized metric spaces

“…Consequently, our results extend, improve and generalize several results in the literature. Our results generalize the results of Chen and Sun [5], Lakzian and Samet [19], Shatanawi et al [21] as well as, our results generalize several well-known comparable results in the literature. To illustrate our results, we conclude the paper with examples.…”

“…Very recently, Chen and Sun [5] improve the Theorem 1.5 by using the notion of weaker (φ, ϕ)-contractive mapping and prove the periodic points and fixed points for this type of contraction. Shatanawi et al [21] also extended the Theorem 1.5 by replacing in (1) the term d(x, y) by the quantity max{d(x, y), d(x, Tx), d(y, Ty)} and the continuity of φ by lower semi-continuity; see also [3]. In the following lemma Z. Kadeburg, S. Radenovic [23] show that the assumption of Hausdorff in the most of fixed point theorems in generalized metric spaces can be omitted.…”

On common fixed points for α−F -contractions and applications

“…Then, several interesting and valuable results appeared in this direction, for example see [2,6,17,20,[23][24][25]. Definition 2.13.…”

Fixed Point Results for Dualistic Contractions with an Application

“…e recent study in the metric xed point theory is due to the Banach Contraction Principle, which has been modi ed, improved, and generalized in many ways in metric spaces (see for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]).…”